Microeconomics II (PhD) Tutorial 3, April 29
Andreas Kleiner akleiner@uni-bonn.de
Exercises:
1. Solve Exercise 23.D.6 in MWG.
2. (Variant of Exercise 23.D.5 in MWG)
Consider asealed-bid all-pay auction in which every buyer submits a bid, the highest bidder receives the good (with symmetric tie-breaking), andevery buyer pays the seller the amount of his bid regardless of whether he wins.
Suppose there are I symmetric buyers and each buyer’s valuation is independently drawn from the interval [θ, θ]⊂R+ according to a stricly positive density.
(a) Argue that any symmetric pure strategy equilibrium of this auction yields the seller the same expected revenue as the sealed-bid second-price auction.
(b) Show that this auction indeed has a symmetric equilibrium in pure strategies.
[Hint: Try to construct an explicit equilibrium using the payoff equivalence result.]
You can use the following envelope theorem in the next exercise. We will discuss the idea of the theorem and its proof in the tutorial.
Theorem (Milgrom and Segal, 2002).
Let X be an arbitrary set,T = [t, t],1 andf :X×T →R. Denote V(t) = sup
x∈X
f(x, t) (1)
X∗(t) ={x∈X|f(x, t) =V(t)}. (2)
Suppose that f(x,·) is differentiable for all x ∈ X, ft(x,·) is uniformly bounded and that X∗(t) 6= ∅ for almost allt. Then for any selectionx∗(t)∈X∗(t),
V(t) =V(t) + Z t
t
ft(x∗(s), s)ds. (3)
3. (advanced)Consider the general mechanism design setting from the lecture, wherevi(k, θi) denotes the value of allocationkto agent iwith typeθi. Suppose that Θi= [θi, θi]⊂Rand thatviis differentiable inθi for all kand the derivative is uniformly bounded. Given a direct revelation mechanism (k, t), let Ui(θ) = vi(k(θ), θi) +ti(θ) be the utility of agent i if θ is the profile of types and all agents report truthfully.
(a) Show that if the direct revelation mechanism (k, t) is implementable in dominant strategies, then
Ui(θ) =Ui(θi, θ−i) + Z θi
θi
∂vi(k(s, θ−i), s)
∂θi ds. (ICFOC)
Suppose that vi(k, θi) has the single-crossing property: ∂2∂k ∂θvi(k,θi)
i exists and is stricly positive for all k∈K andθi∈Θi.
(b) Show that if the direct revelation mechanism (k, t) is implementable in dominant strategies, then k(θi, θ−i) is weakly increasing inθi for allθ−i.
(c) Show that any monotone mechanism that satisfies (ICFOC) is implementable in dominant strategies.
(d) Discuss the relation of these results to the result that you saw in the lecture.
(e) Show: If a direct revelation mechanism implements the value-maximizing allocation rule in dominant strategies, then it is a VCG mechanism.
1This result holds more generally, for example ifT⊂Rnis convex.
